Ethic Duality: A Homological Framework for Primal-Dual Problems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Pasechnyuk-Vilensky, Dmitry Takáč, Martin |
| author_facet | Pasechnyuk-Vilensky, Dmitry Takáč, Martin |
| contents | We develop a homological duality framework based on a contravariant functor $D=\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)η=ηf$. In the derived setting, the functor $\mathrm{RHom}_E(-,R)$ produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer $\operatorname{Ext}^1$ identifies primal-dual gaps, while higher $\operatorname{Ext}^k$ provide a systematic hierarchy of derived obstructions to exactness.
This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of $\operatorname{Ext}^1$, and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as instances of ethic exactness. In dynamical systems, the higher derived layers encode nonvanishing persistence phenomena. Additional examples include social-choice configurations, categorical factorization in scattering formalisms, coding-theoretic duality, and Bellman-type recurrences, all appearing as concrete instances of Ext-controlled exactness.
All resulting invariants are stable under derived Morita equivalence and depend only on the dualizing pair $(E,R)$. The framework therefore provides a substrate-independent criterion for primal-dual exactness and a uniform homological description of its obstructions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ethic Duality: A Homological Framework for Primal-Dual Problems Pasechnyuk-Vilensky, Dmitry Takáč, Martin Category Theory Optimization and Control 18G10, 90C25, 68Q17 F.1.1; G.1.6; F.2.2 We develop a homological duality framework based on a contravariant functor $D=\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)η=ηf$. In the derived setting, the functor $\mathrm{RHom}_E(-,R)$ produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer $\operatorname{Ext}^1$ identifies primal-dual gaps, while higher $\operatorname{Ext}^k$ provide a systematic hierarchy of derived obstructions to exactness. This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of $\operatorname{Ext}^1$, and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as instances of ethic exactness. In dynamical systems, the higher derived layers encode nonvanishing persistence phenomena. Additional examples include social-choice configurations, categorical factorization in scattering formalisms, coding-theoretic duality, and Bellman-type recurrences, all appearing as concrete instances of Ext-controlled exactness. All resulting invariants are stable under derived Morita equivalence and depend only on the dualizing pair $(E,R)$. The framework therefore provides a substrate-independent criterion for primal-dual exactness and a uniform homological description of its obstructions. |
| title | Ethic Duality: A Homological Framework for Primal-Dual Problems |
| topic | Category Theory Optimization and Control 18G10, 90C25, 68Q17 F.1.1; G.1.6; F.2.2 |
| url | https://arxiv.org/abs/2512.17170 |